Ramification in Iwasawa theory and splitting conjectures (Q2878684)

Motivated by analogies of curves over finite fields the authors of the paper under review formulate a `reciprocity conjecture' in Iwaswa theory. Moreover, they state certain `splitting conjectures' which are shown to be equivalent to Leopoldt's conjecture.NEWLINENEWLINELet \(X\) be a smooth projective curve over a field \(K\) and choose two points \(P,Q \in X(K)\). Let \(X_{P,Q}\) be the singular curve obtained from \(X\) by identifying \(P\) with \(Q\). We then have an exact sequence NEWLINE\[NEWLINE0 \rightarrow \mathbb G_m \rightarrow J_{P,Q} \rightarrow J \rightarrow 0,NEWLINE\]NEWLINE where \(J_{P,Q}\) and \(J\) denote the (generalized) Jacobians of \(X_{P,Q}\) and \(X\), respectively. Then the class of this sequence in \(\mathrm{Ext}^1(J, \mathbb G_m)\) is given by the degree zero divisor \((Q) - (P)\) under \(\mathrm{Ext}^1(J, \mathbb G_m) \simeq \mathrm{Pic}^0(J) = J\).NEWLINENEWLINEThe authors introduce the following analogue in Iwasawa theory: Let \(F\) be a totally real field and \(p\) and odd prime. Let \(F_{\infty}\) be the cyclotomic \(\mathbb Z_p\)-extension of \(F\) and let \(M_{\infty}\) be the maximal abelian pro-\(p\)-extension of \(F_{\infty}\) unramified outside \(p\). Then \(Y_{\infty} := \mathrm{Gal}(M_{\infty} / F_{\infty})\) is a finitely generated torsion \(\Lambda\)-module, where as usual \(\Lambda = \mathbb Z_p[[\Gamma]] \simeq \mathbb Z_p[[T]]\), \(\Gamma = \mathrm{Gal}(F_{\infty}/F)\). Let \(Y_{\infty}' := \mathrm{Gal}(M_{\infty} / F)\) and consider the exact sequence NEWLINE\[NEWLINE0 \rightarrow Y_{\infty} \rightarrow Y_{\infty}' \rightarrow \mathbb Z_p \rightarrow 0.NEWLINE\]NEWLINE The authors call the last map a `degree map'. From this one obtains NEWLINE\[NEWLINE0 \rightarrow (Y_{\infty})_{\Gamma} \rightarrow (Y_{\infty}')_{\Gamma} \rightarrow \mathbb Z_p \rightarrow 0.NEWLINE\]NEWLINE Now fix a finite set \(Q\) of places of \(F\) away from \(p\). Define \(M_Q'\) to be the \(\mathbb Z_p\)-submodule of \((Y_{\infty}')_{\Gamma}\) generated by the Frobenii at \(q\) for \(q \in Q\). Let \(M_Q\) be the submodule of \(M_Q'\) of those elements which have degree zero.NEWLINENEWLINEWhen \(Q\) consists of two points, the authors associate a class in \(H^1(\Gamma, Y_{\infty})\) and denote the \(\mathbb Z_p\)-submodule of \(H^1(\Gamma, Y_{\infty})\) generated by this class by \(N_Q\). The analogue of the above is the following `reciprocity conjecture': \(N_Q\) is mapped isomorphically onto \(M_Q\) under \(H^1(\Gamma, Y_{\infty}) \simeq (Y_{\infty})_{\Gamma}\).NEWLINENEWLINEThis has meanwhile been shown by \textit{R. T. Sharifi} [Int. Math. Res. Not. 2014, No. 5, 1409--1424 (2014; Zbl 1307.11122)].NEWLINENEWLINENow put \(\mathcal F_{\infty} := F(\mu_{p^{\infty}})\) and let \(L_Q\) be the maximal abelian pro-\(p\)-extension of \(\mathcal F_{\infty}\) that is unramified outside \(Q\). Put \(X_{\infty,Q} := \mathrm{Gal}(L_Q / \mathcal F_{\infty})\) which has an action by complex conjugation. Then there is an exact sequence NEWLINE\[NEWLINE0 \rightarrow \mathbb Z_p(1)^{m-1} \rightarrow X_{\infty,Q}^- \rightarrow X_{\infty,\emptyset}^- \rightarrow 0,NEWLINE\]NEWLINE where \(m\) is the cardinality of \(Q\). It is shown that Leopodt's conjecture (at \(p\)) is true if and only if this sequence splits up to isogeny (i.e.~after tensoring with \(\mathbb Q_p\)) for all \(Q = \left\{q_1, q_2\right\}\) such that \(q_1\) and \(q_2\) are inert in \(F_{\infty} / F\).NEWLINENEWLINEFinally, there is a variant of the splitting conjecture, and the authors show that for a totally real Galois extension of \(\mathbb Q\) the Leopoldt defect is not equal to \(1\).